Definition

Suppose is a group that has at least one minimal generating set, i.e., a generating set with the property that no proper subset of it generates . The maximum size of minimal generating set for is defined as the maximum of the sizes of all possible minimal generating sets for .

Note that minimal generating sets are also called irredundant generating sets and hence the maximum size of minimal generating set can also be called the maximum size of irredundant generating set.

Infinite, finitely generated groups

In the case that is infinite but finitely generated, it possesses finite minimal generating sets, and in fact, every minimal generating set is finite (see equivalence of definitions of finitely generated group). However, there may or may not be a finite maximum size of minimal generating set. Examples of both kinds are below:

For the group of integers, there is no upper bound on the maximum size of a minimal generating set. Specifically, for any positive integer , we can take distinct primes, and consider a generating set that comprises products of these primes, at a time. This is a minimal generating set.

There do exist infinite groups where there is a maximum size of minimal generating set. For instance, the Tarski groups are infinite groups with the property that any two elements that are not redundant with each other generate the whole group.